For anyone interested, the picture Figure 1 [0] - as one would except - is quite messy but yeah, Equus (most recent common ancestor) node 1 and node 2 (3.9 - 7.8 mya (~5.6 mya)) and the divergence (node 3) to caballine/non-caballine Equus @ ~4.0–4.5 mya; those bifurcations - based on the found fossils - all originated from North America (orange color).

[next @~1mya (the lower orange (North America) caballine arm from node 3): leading to E. caballus (green: Eurasia) --> domesticated [1] horses]

Regarding the causes of the Late Quaternary extinctions [2] it is pretty hard to pinpoint a single cause e.g. climate change etc. and there are a lot of hypotheses floating around (probably the most curious/spectacular of the bunch: younger dryas impact theory).

But at some point regardless of the causes - approx. 10.000 years ago - there are no traces found of (wild) horses in the Americas, anymore; they only reemerge again with the reintroduction by the European colonists in the 15/16th century.

[0]https://elifesciences.org/articles/29944/figures#content

[1]https://www.nationalgeographic.com/science/article/horse-dom...

[2]https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8654998/#!po=32...